Let $A$ be a $3\times 3$ matrix such that $A^3=0$. Which of the following is always true?

Solution

1. Since $A^3=0$, $A$ is a nilpotent matrix.
2. The eigenvalues of a nilpotent matrix are all $0$.
3. The eigenvalues of $I+A$ are $1 + 0 = 1$.
4. The determinant is the product of eigenvalues, so $$\begin{array}{rl}\det (I+A)& =1\times 1\times 1\\ & =1\end{array}$$. Hence the answer is (B).